More accurate formula for the Chebyshev $\psi$ function

Question

In the standard books of Analytic Number Theory we see the following formula for $\psi$ Chebyshev:

$$ \psi(x)=\sum_{n\le x} \Lambda(n) = \frac 1 {2\pi i} \int_{b-iT}^{b+iT} \left( -\frac{\zeta'(s)}{\zeta(s)} \right) \frac{x^s} s \, ds + O\left( \frac{x\ln^2 x} T \right) $$

Problem: It is known that we can obtain more accurate formula by changing $O(\frac{x\ln^2 x}{T})$ with $O(\frac{x}{T}(\ln x) \times (\ln \ln x))+O(\ln x)$. Do you know its proof or a reference to find its proof? Thank you for your help!

Answer 1

The original result is due to Goldston "On a result of Littlewood concerning prime numbers. II".

This result has recently been made explicit, see (https://arxiv.org/abs/2107.14468v1).

Ramaré also has other, asymptotically stronger results available here (https://ramare-olivier.github.io/Maths/TruncatedPerron-5.pdf). This work has been published.

In particular, Ramaré's Theorem 1.1 gives an O(xlogx/T) error term under reasonable conditions. And, Theorem 1.2 gives an O(x/T) error term if the truncation is not sharp. However, one should note that there is a small error in the result of Ramaré's Theorem 1.2. See the addendum (https://ramare-olivier.github.io/Maths/TruncatedPerron-5-Addendum-04.pdf) for some discussion of this.